TERMINAL SIDE: lines & trigonometric functions

Question

Find sin (angle) if the terminal side of (angle) is in the third quadrant on the line y=1/2x

Answer 1

Hi, Tumaiesla

 

Here the  thinking process and the solution.

 

Sketch a graph of the equation y = 1/2x by plotting the points (2,1) and (-2,-1) and draw the line through them.

 

Step 1 - Draw the sides of the right triangle that go from the origin (0,0) to (2,0) and the from (2,0) up to (2,1).

The side of the triangle on the x-axis has length two and the other side has length one.

 

Step 2 - Use the Pythagorean theorem to find the length of the hypotenuse:

c = sqrt(2² + 1²) = sqrt(5) The sin of the angle opposite the side of length 1 is sin(1/sqrt(5)) = 0.4472135955

 

Step 3 - Find the angle. If we compute arcsin(0.4472135955) this gives us 26.56505118 degrees.

 

Step 4 - The angle you want is 180 degrees more than this one so add 180 to 26.56505118 = 206.56505118

 

Step 5 - and take the sin of it. sin (206.56505118) = -0.4472135955 and you round to whatever your teacher asked for. And that's the sin you're looking for.

 

However, the first sin we got is the opposite of the answer.

 

This is because sin(180 + a) = -sin(a). I wasn't sure if you knew this, if you had you could have skipped the steps 4 and 5 and gone right to the answer:

 

sin(180+26.56505118) = -sin(26.56505118) = -0.4472135955

 

This would be much easier to explain with an actual graph. I tried to paste one to this e-mail, but it wouldn't let me.